Show that f in example 3.1 is linear. The function f : 2 2 defined by

Show that f in example 3.1 is linear.
The function f : ℜ2 → ℜ2 defined by
F( x1, x2) = (x1 cos θ - x2 sin θ ,x1 sin θ + x2 cos θ), 0 ≤ 0 2π
rotates any vector in the plane counter clock wise through the angle θ (figure 2.3). It is easily verified that f is linear. Linearity implies that rotating the sum of two vectors yields the same result as summing the rotated vectors.